How did the Mongol invasions affect the relationship between China and Japan?
Answers
The effect was somewhere from somewhat important to absolutely vital. There is disagreement in the field, but it is generally accepted that the Mongols were an existential threat that hardened the Shogunate's control of the Honshu-Kyushu-Shikoku regions. This threat began in the 1220's when Korea fell under mongol control. The Kamakura shogunate was the first truly powerful shogunate, and represented the fundamental shift in Japan's political center away from the Heian region (modern Kyoto) to Kamakura (modern Tokyo).
The Kamakura set the mold for power in the core of Japan, and that mold was fired in the crucible that was the existential threat of Mongol invasion.
Even if the wars with the mongols were not as major as japanese sources suggest, the sources can be read in another way, as propaganda. Which again, supports the idea that the mongol threat was formative - as it was a real or mythologized story that encouraged greater central control and power